Coherent Sheaves Research Articles

Representation Theory of Geigle-Lenzing Complete Intersections

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings R R graded by abelian groups L \mathbb {L} of rank 1 1 , which we call Geigle-Lenzing complete intersections. We study the stable category C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R of Cohen-Macaulay representations, which coincides with the singularity category D s g L ( R ) \mathsf {D}^{\mathbb {L}}_{\mathrm {sg}}(R) . We show that C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R is triangle equivalent to D b ( m o d A C M ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{\mathrm {CM}}) for a finite dimensional algebra A C M A^{\mathrm {CM}} , which we call the CM-canonical algebra. As an application, we classify the ( R , L ) (R,\mathbb {L}) that are Cohen-Macaulay finite. We also give sufficient conditions for ( R , L ) (R,\mathbb {L}) to be d d -Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category c o h X = m o d L R / m o d 0 L R \mathsf {coh}\mathbb {X}=\mathsf {mod}^{\mathbb {L}}R/\mathsf {mod}^{\mathbb {L}}_0R of coherent sheaves on the Geigle-Lenzing projective space X \mathbb {X} . Geometrically this is the quotient stack X = [ X / G ] \mathbb {X}=[X/G] for X = S p e c R ∖ { R + } X={\mathrm {Spec}}\,R\setminus \{R_+\} and G = S p e c k [ L ] G={\mathrm {Spec}}\,k[\mathbb {L}] . We show that D b ( c o h X ) \mathsf {D}^{\mathrm {b}}(\mathsf {coh}\mathbb {X}) is triangle equivalent to D b ( m o d A c a ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{ca}) for a finite dimensional algebra A c a A^{\mathrm {ca}} , which we call a d d -canonical algebra. We study when X \mathbb {X} is d d -vector bundle finite, and when X \mathbb {X} is derived equivale’nt to a d d -representation infinite algebra in the sense of higher Auslander-Reiten theory. Our d d -canonical algebras provide a rich source of d d -Fano and d d -anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of D s g L ( R ) \mathsf {D}_{\mathrm {sg}}^{\mathbb {L}}(R) and D b ( c o h X ) \mathsf {D}^{b}(\mathsf {coh}\mathbb {X}) .

Admissible subcategories of del Pezzo surfaces

We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivial admissible subcategories, but provably do not contain phantoms. We also prove that any admissible subcategory supported set-theoretically on a smooth (−1)-curve in a surface is generated by some twist of the structure sheaf of that curve.

Chow transformation of coherent sheaves

Abstract We define a dual of the Chow transformation of currents on any complex projective manifold. This integral transformation is a factor of a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear differential operator, which does not commute with ∂ \partial or ∂ ¯ \overline{\partial } . We obtain a complete intrinsic resolution of the problem of the algebraicity of the cohomology classes. On another hand, in the case of the complex projective space, we give the translation in terms of real-analytic D {\mathcal{D}} -modules of the properties of the Chow transformation. Then, the proofs can be simplified by using the conormal currents, which exist for all currents of bidimension ( p , p ) \left(p,p) on the complex projective space, even not closed. This is a consequence of the existence of dual currents, defined on the dual complex projective space. In particular, we obtain a linear differential system of order lower than that of the Gelfand-Gindikin-Graev differential system, characterizing the images by the Chow transformation of smooth differential forms on the complex projective space.

Stably semiorthogonally indecomposable varieties

A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI, then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are induced from decompositions of $X$. We prove that any variety whose Albanese morphism is finite is NSSI, and that the total space of a fibration over NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces $C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive genus.

Homological mirror symmetry for the symmetric squares of punctured spheres

For an appropriate choice of a Z-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a (k+3)-punctured sphere, i.e. the Weinstein manifold given as the complement of (k+3) generic lines in CP2 is quasi-equivalent to the derived category of coherent sheaves on a singular surface Z2,k constructed as the boundary of a toric Landau-Ginzburg model (X2,k,w2,k). We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localizing. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models (Xn,k,wn,k) are constructed from the combinatorics of curves on the punctured sphere and are related to small toric resolutions of the singularity x1…xn+1=v1…vk.

Mapping stacks and categorical notions of properness

One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong $h$-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

Admissible Homomorphisms and Equivariant Relations Between Weighted Projective Lines

Abstract The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant relation is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant relations for the weighted projective lines of domestic and tubular types, respectively.

Coherent sheaves, Chern classes, and superconnections on compact complex-analytic manifolds

A twist-closed enhancement of the bounded derived category $\mathcal{D}^b_{\mathrm{coh}} (X)$ of complexes of $\mathcal{O}_X$-modules with coherent cohomology is constructed by means of the DG-category of $\overline\partial$-superconnections. The machinery of $\overline\partial$-superconnections is applied to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$-folds

Decompositions of moduli spaces of vector bundles and graph potentials

Abstract We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

Stability conditions for polarised varieties

Abstract We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity

We give a generating function for the number of pairs of n×n matrices (A,B) over a finite field that are mutually annihilating, namely, AB=BA=0. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over C[u,v]/(uv) as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.

Exotic t-structures for two-block Springer fibres

We study the category of representations of s l m + 2 n \mathfrak {sl}_{m+2n} in positive characteristic, with p p -character given by a nilpotent with Jordan type ( m + n , n ) (m+n,n) . Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to D m , n 0 \mathcal {D}_{m,n}^0 , the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories D m , n \mathcal {D}_{m,n} (i.e. for different values of n n ). This allows us to describe the irreducible objects in D m , n 0 \mathcal {D}_{m,n}^0 and enumerate them by crossingless ( m , m + 2 n ) (m,m+2n) matchings. We compute the E x t \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov’s arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category O \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al.

A note on Bridgeland stability conditions and Catalan numbers

In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its autoequivalence group on its Bridgeland stability manifold. In solving an equation involving this group action, the generating function of Catalan numbers arises, allowing us to use asymptotic estimates of Catalan numbers to arrive at a bound for the solution set.

Derived equivalence and fibrations over curves and surfaces

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus g≥2. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base dimension under the condition that all the Hodge numbers of type h0,p(X) are derived invariant.

Two-dimensional categorified Hall algebras

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathcal{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathcal{M}$. In the surface case, $\mathbb{R}\mathcal{M}$ is a suitable derived enhancement of the moduli stack $\mathcal{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov–Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala–Schiffmann, while in the other two cases our construction yields, by passing to $\mathsf{K}0$, new K-theoretical Hall algebras, and by passing to $\mathsf{H}\ast^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann–Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve.

O-minimal GAGA and a conjecture of Griffiths

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

Theı\imathHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of11-periodic complexes of coherent sheaves on the projective line. Thisı\imathHall algebra is shown to realize the universalqq-Onsager algebra (i.e.,ı\imathquantum group of split affineA1A_1type) in its Drinfeld type presentation. Theı\imathHall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these twoı\imathHall algebras, explaining the isomorphism of theqq-Onsager algebra under the two presentations.